Certainly! Let's determine the residuals to plot on our residual plot step by step.
First, for each data point [tex]\((x, y)\)[/tex], we will find the value of [tex]\(y\)[/tex] predicted by the line of best fit, [tex]\(y_fit\)[/tex], using the equation [tex]\( y = -0.33x + 4.27 \)[/tex]. Then, we will calculate the residuals, which are defined as the difference between the observed [tex]\(y\)[/tex] value and the predicted [tex]\(y_fit\)[/tex] value.
[tex]\[
\text{Residual} = y - y_{\text{fit}}
\][/tex]
Let's proceed with each data point:
1. For [tex]\((1, 4.0)\)[/tex]:
- Calculate [tex]\(y_{\text{fit}}\)[/tex]: [tex]\(y_{\text{fit}} = -0.33(1) + 4.27 = 3.94\)[/tex]
- Calculate the residual: Residual [tex]\(= 4.0 - 3.94 = 0.06\)[/tex]
2. For [tex]\((2, 3.3)\)[/tex]:
- Calculate [tex]\(y_{\text{fit}}\)[/tex]: [tex]\(y_{\text{fit}} = -0.33(2) + 4.27 = 3.61\)[/tex]
- Calculate the residual: Residual [tex]\(= 3.3 - 3.61 = -0.31\)[/tex]
3. For [tex]\((3, 3.8)\)[/tex]:
- Calculate [tex]\(y_{\text{fit}}\)[/tex]: [tex]\(y_{\text{fit}} = -0.33(3) + 4.27 = 3.28\)[/tex]
- Calculate the residual: Residual [tex]\(= 3.8 - 3.28 = 0.52\)[/tex]
4. For [tex]\((4, 2.6)\)[/tex]:
- Calculate [tex]\(y_{\text{fit}}\)[/tex]: [tex]\(y_{\text{fit}} = -0.33(4) + 4.27 = 2.95\)[/tex]
- Calculate the residual: Residual [tex]\(= 2.6 - 2.95 = -0.35\)[/tex]
5. For [tex]\((5, 2.7)\)[/tex]:
- Calculate [tex]\(y_{\text{fit}}\)[/tex]: [tex]\(y_{\text{fit}} = -0.33(5) + 4.27 = 2.62\)[/tex]
- Calculate the residual: Residual [tex]\(= 2.7 - 2.62 = 0.08\)[/tex]
Summarizing, the residuals for the given data points are:
[tex]\[
[(1, 0.06), (2, -0.31), (3, 0.52), (4, -0.35), (5, 0.08)]
\][/tex]
Therefore, the correct residual plot would consist of the points:
- [tex]\((1, 0.06)\)[/tex]
- [tex]\((2, -0.31)\)[/tex]
- [tex]\((3, 0.52)\)[/tex]
- [tex]\((4, -0.35)\)[/tex]
- [tex]\((5, 0.08)\)[/tex]
These residuals capture how far each actual [tex]\(y\)[/tex] value is from the predicted [tex]\(y\)[/tex] value given by the line of best fit for each [tex]\(x\)[/tex] value.
